Dr. Vidarte-Olivera, Jhon

We investigate the qualitative characteristics of a test particle attracted to an irregular elongated body, modeled as a non-homogeneous straight segment with a variable linear density. By deriving the potential function in closed form, we formulate the Hamiltonian equations of motion for this system. Our analysis reveals a family of periodic circular orbits parameterized by angular momentum. Additionally, we utilize the axial symmetry resulting from rotations around the segment’s axis to consider the corresponding reduced system. This approach identifies several reduced-periodic orbits by analyzing appropriate Poincaré sections. These periodic orbits are then reconstructed into quasi-periodic orbits within the full dynamical system.

This work investigates the qualitative dynamics of a massless particle around elongated bodies modeled with a linearly varying density function. We assume that the segment rotates uniformly with angular velocity w, with the fixed segment considered as a special case. For the rotating straight segment, we identify four equilibria: two collinear and two triangular. The primary novelty of our work is the detection of a pitchfork bifurcation, wherein, under slow and moderate rotation, the triangular equilibria shift toward and merge with one of the collinear points, while remaining distinct in fast rotations. This feature, previously unreported, is visualized through Hill’s regions and highlights how both rotation rate and asymmetry in mass distribution affect orbital dynamics, providing new insights into gravitational environments near irregularly shaped bodies. Additionally, we analyze the linear stability of the relative equilibria. Finally, for the fixed straight segment, we examine singular solutions, showing that any solution not defined globally over time leads the particle to eventually approach the segment at a distance of zero. In the one-dimensional problem, all singularities are observed to result from collisions.

We analyse the dynamics of an infinitesimal particle around an elongated body, which is modelled as a homogeneous fixed straight segment centred at the origin. We assume that the length of the segment is small compared with the distance to the particle. After a Lie–Deprit normalization, we end up with a Hamiltonian that has not only the mean anomaly but also the argument of the perigee relegated to terms or third order or higher. We employ invariant and reduction theories to reduce the artificial symmetries associated with the Kepler flow and the central action of the angular momentum. Analysing the relative equilibria in the first and second reduced spaces allows us to determine the existence of near-polar circular periodic orbits and KAM tori.